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Chapter 5: Problem 80

A telephone pole is 55 feet tall. A guy wire 80 feet long is attached from the ground to the top of the pole. Find the angle between the wire and the pole to the nearest degree.

Short Answer

Expert verified
The angle between the wire and the pole is approximately 45 degrees to the nearest degree.

Step by step solution

01

Understand the problem context

Visualize or sketch the problem. The telephone pole, the ground, and the guy wire form a right-angled triangle, with the wire serving as the hypotenuse.
02

Identify the trigonometric ratio

Identify the trigonometric ratio to help solve the problem. In this case, the tangent of the angle formed by the pole and the wire can be used because it is the ratio of the opposite side (height of the pole) to the adjacent side (distance from the base). Thus, \(\tan \theta = \frac{\text{opposite}}{adjacent}\)
03

Substitute the given values

Substitute the given values into the equation. The height of the pole is 55 feet, and the length of the guy wire is 80 feet, which forms the hypotenuse of the triangle. Use the Pythagorean theorem to calculate the ground distance, which is \( \sqrt{80^{2} - 55^{2}} = 53.64 \) feet, approximately. Thus, \(\tan \theta = \frac{55}{53.64}\)
04

Calculate the angle

Apply the inverse tangent function to both sides of the equation to calculate the angle, 胃. Thus, \(\theta = \tan^{-1} (\frac{55}{53.64} ) \)
05

Find the value of angle

Use a scientific calculator to find the value. After calculating, the value of the angle would be approximately 45 degrees to the nearest degree

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